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\begin{document}

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\section*{curl}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{riemannian_geometry}{}\paragraph*{{Riemannian geometry}}\label{riemannian_geometry}

[[!include Riemannian geometry - contents]]

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{Definition}{Definitions}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{InTermsOfDifferentialForms}{In terms of differential forms}\dotfill \pageref*{InTermsOfDifferentialForms} \linebreak
\noindent\hyperlink{ViaIntegration}{Via integration}\dotfill \pageref*{ViaIntegration} \linebreak
\noindent\hyperlink{via_cross_products}{Via cross products}\dotfill \pageref*{via_cross_products} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{Stokes}{Relation to the Stokes theorems}\dotfill \pageref*{Stokes} \linebreak
\noindent\hyperlink{remark}{Remark}\dotfill \pageref*{remark} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{Definition}{}\subsection*{{Definitions}}\label{Definition}

\hypertarget{InTermsOfDifferentialForms}{}\subsubsection*{{In terms of differential forms}}\label{InTermsOfDifferentialForms}

In [[Riemannian geometry]], the \textbf{curl} or \textbf{rotation} of a [[vector field]] $v$ on an [[orientation|oriented]] $3$-[[dimension|dimensional]] [[Riemannian manifold]] $(X,g)$ is the vector field denoted $curl(v)$ (or $rot(v)$ or $\Del \times v$) defined by

\begin{displaymath}
curl(v) \;\coloneqq\; g^{-1} \left(\star_g d_{dR}\, g(v) \right) 
  \,.
\end{displaymath}
where

\begin{enumerate}%
\item $\Gamma(T X) \underoverset{\underset{g}{\longrightarrow}}{\overset{g^{-1}}{\longleftarrow}}{\phantom{AA}\simeq\phantom{AA}} \Omega^1(X)$ is the [[linear isomorphism]] between [[vector fields]] and [[differential 1-forms]] given by the [[metric tensor]] $g$;


\item $d_{dR} \;\colon\; \Omega^n(X) \longrightarrow \Omega^{n+1}(X)$ is the [[de Rham differential]]


\item $\star_g \;\colon\; \Omega^n(X) \to \Omega^{dim(X)-n}(X)$ is the [[Hodge star operator]] (which uses the orientation of $X$).



\end{enumerate}
Notice that for this to make sense it is crucial that the [[dimension]] of $X$ is $3$, for only then is the Hodge dual of the de Rham differential of a $1$-form again a $1$-form; that is, $n = 3$ is the unique solution of $n - (1 + 1) = 1$.

\hypertarget{ViaIntegration}{}\subsubsection*{{Via integration}}\label{ViaIntegration}

Alternatively, the curl/rotation of a vector field $\vec v$ at some point $x\in X$ may be defined as the [[limit of a sequence|limit]] [[integral]] formula

\begin{displaymath}
\vec{n}\cdot rot \vec v 
  = 
  \lim_{area S\to 0} 
  \frac{1}{area S} 
  \oint_{\partial S} \vec{t}\cdot \vec v d r
\end{displaymath}
where $D$ runs over the smooth oriented [[surfaces]] ([[submanifolds]] of [[dimension]] $2$) containing the point $x$ and with smooth [[boundary]] $\partial D$, $\vec{n}$ is the unit vector through the surface $S$, and $\vec{t}$ is the unit vector tangent to the [[curve]] $\partial S$. (We use the orientation of $X$ to determine the [[direction of a vector|direction]] of $\vec{n}$ from the [[orientation]] of $S$.)

This formula does not depend on the shape of boundaries taken in limiting process, so one can typically take a [[coordinate chart]] and [[discs]] with decreasing radius in this particular coordinate chart. One can even use $\pi r^2$ in place of the actual area of the disc around $x$ with coordinate-radius $r$, to save on calculating this area, as long as the coordinate chart assigns the standard coordinates to the metric at $x$.

The proof that this definition is coherent and agrees with the previous one is essentially the [[Kelvin–Stokes Theorem]]; see \hyperlink{Stokes}{below} for discussion.

\hypertarget{via_cross_products}{}\subsubsection*{{Via cross products}}\label{via_cross_products}

More generally, if $(X,g)$ is a [[Riemannian manifold]] whose [[cotangent spaces]] (equivalently, [[tangent spaces]]) are smoothly equipped with a binary [[cross product]] $&#10761;\colon \Omega^2(X;\mathbb{R}) \to \Omega^1(X;\mathbb{R})$, then the \textbf{curl} of any vector field $v$ is

\begin{displaymath}
curl(c) 
  \;=\; g^{-1} &#10761; d_{dR} g(v)
\end{displaymath}
However, this is not as general as it may appear:

\begin{itemize}%
\item in $0$ or $1$ [[dimension]], the cross product, hence the curl, must always be $0$;
\item in $3$ dimensions, a smooth choice of cross product is equivalent to a smooth choice of [[orientation]], and we recover the previous formula;
\item in $7$ dimensions, if a smooth choice of cross product is possible (as on the [[7-sphere]]), then uncountably many are possible, giving as many different notions of curl;
\item in any other number of dimensions, no binary cross product exists at all, hence no curl.

\end{itemize}
That said, there are also cross products of other [[arities|arity]] in other dimensions; using essentially the same formula, we can take the curl of a $k$-[[multivector field]] if we have a smooth $(k+1)$-ary cross product. Or if the cross product is other than vector-valued, then we can obtain a curl that is other than a vector field.

In particular, in $2$ dimensions, we have the \textbf{scalar curl}

\begin{displaymath}
curl(c) 
  \;=\; &#10761; d_{dR} g(v)
\end{displaymath}
where $&#10761;\colon \Omega^2(X;\mathbb{R}) \to \Omega^0(X;\mathbb{R})$ is the [[volume form]] (or area form) on the $2$-dimensional Riemannian manifold $X$.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

If $(X,g)$ is $\mathbb{R}^3$ endowed with the canonical Euclidean metric, then the curl of a vector field $(v^1,v^2,v^3) = v^1\partial_1 + v^2\partial_2 + v^3\partial_3$ is

\begin{displaymath}
curl(v)^1 = \frac{\partial v^3}{\partial x^2}-\frac{\partial v^2}{\partial x^3} 
;\qquad
curl(v)^2 = \frac{\partial v^1}{\partial x^3}-\frac{\partial v^3}{\partial x^1} 
;\qquad
curl(v)^3 = \frac{\partial v^2}{\partial x^1}-\frac{\partial v^1}{\partial x^2} 
.
\end{displaymath}
This is the classical curl from [[vector analysis]].

If $(X,g)$ is $\mathbb{R}^2$ endowed with the canonical Euclidean metric, then the curl of a vector field $(v^1,v^2) = v^1\partial_1 + v^2\partial_2$ is

\begin{displaymath}
curl(v) = \frac{\partial v^2}{\partial x^1}-\frac{\partial v^1}{\partial x^2} 
.
\end{displaymath}
\hypertarget{Stokes}{}\subsection*{{Relation to the Stokes theorems}}\label{Stokes}

Recall that if $X$ is an $n$-dimensional [[differentiable manifold]], $D$ is a $p$-dimensional [[submanifold]] [[manifold with boundary|with boundary]], and $\alpha$ is a [[differentiable function|differentiable]] $(p-1)$-rank [[exterior differential form]] on a [[neighbourhood]] of $D$ in $X$, then the generalized [[Stokes Theorem]] says that the [[integration of differential forms|integral]] of $\alpha$ on the boundary $\partial{D}$ equals the integral on $S$ of the [[de Rham differential]] $\mathrm{d}_{DR}\alpha$.

When $n=3$, $p=2$, and $X$ is equipped with an orientation and a metric, then this is equivalent to saying that the integral of a vector field $v$ along the boundary of an oriented surface $D$ in $X$ is equal to the integral of the vector field's curl across the surface:

\begin{equation}
\int_{\partial{D}} v \cdot \mathrm{d}\mathbf{r} = \int_D curl v \cdot \mathrm{d}\mathbf{S} .
\label{KS}\end{equation}
(In particular, when $X$ is $\mathbb{R}^3$ with its standard orientation and metric, then this is equivalent to the classical Kelvin--Stokes Theorem.) At least, this is what it says if the curl is defined \hyperlink{InTermsOfDifferentialForms}{in terms of differential forms}; if the curl is defined \hyperlink{ViaIntegration}{via integration} instead, then \eqref{KS} is immediate, and the Kelvin--Stokes Theorem says that this definition matches the other one.

When $n=2$, $p=2$, and $X$ is equipped with an orientation and a metric, then the Stokes Theorem is equivalent to saying that the integral of a vector field $v$ along a [[simple closed curve]] in $X$ is equal to the integral of the vector field's scalar curl on the region $D$ enclosed by the curve:

\begin{equation}
\int_{\partial{D}} v \cdot \mathrm{d}\mathbf{r} = \int_D curl v \mathrm{d}A .
\label{Green}\end{equation}
(In particular, when $X$ is $\mathbb{R}^2$ with its standard orientation and metric, then this is equivalent to the curl-circulation form of [[Green's Theorem]].)

\hypertarget{remark}{}\subsection*{{Remark}}\label{remark}

In many classical applications of the curl in [[vector analysis]], the Riemannian structure is actually irrelevant, and the gradient can be replaced with the [[exterior differential|deRham differential]] $d_{dR}$. That is, $X$ is treated as the $1$-form $g(X)$, its curl is treated as the $2$-form $d_{dR} g(X)$, and once these identifications are made there is no need to involve $g$ or $X$ directly at all.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[nabla]]


\item [[gradient]]

\begin{itemize}%
\item [[gradient flow]]


\item [[symplectic gradient]]



\end{itemize}

\item [[Hamiltonian flow]]


\item \textbf{curl}


\item [[divergence]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Curl_%28mathematics%29}{Curl}}

\end{itemize}
[[!redirects curl]] [[!redirects curls]] [[!redirects curl of a vector field]] [[!redirects curls of vector fields]] [[!redirects rotation of a vector field]] [[!redirects rotations of vector fields]]



\end{document}